On a non-Muslim spouse converting to Islam while married to a non-Muslim:

The marriage would be dissolved between non-Muslim couple, if one of them embrace Islam. However, if the other spouse also convert to Islam during the period of iddah (waiting period of three months after divorce), their marriage is still intact.

In a situation where one of the non-Muslim couple intend to embrace Islam, they should be explained about the dissolution of marriage that would take place due to the conversion if one of them refuse to do that. A wise , endearing way should be applied in order to persuade them to embrace Islam.

[Note: For alternate opinions see news articles under the keyword: conversion.]